DORAL, Fla. — Donald J. Trump said on Wednesday that he hoped Russian intelligence services had successfully hacked Hillary Clinton’s email, and encouraged them to publish whatever they may have stolen, essentially urging a foreign adversary to conduct cyberespionage against a former secretary of state.

“Russia, if you’re listening, I hope you’re able to find the 30,000 emails that are missing,” Mr. Trump said during a news conference here in an apparent reference to Mrs. Clinton’s deleted emails. “I think you will probably be rewarded mightily by our press.”

His comments came amid questions about the hacking of the Democratic National Committee’s computer servers, which American intelligence agencies have told the White House they have “high confidence” was the work of the Russian government.

At the same news conference, Mr. Trump also appeared to leave the door open to accepting Russia’s annexation of Crimea two years ago — which the United States and its European allies consider an illegal seizure of territory. That seizure, and the continued efforts of Russian-aided insurgents to undermine the government of Ukraine, are the reason that the United States and its allies still have economic sanctions in force against Moscow.

When asked whether he would recognize Crimea “as Russian territory” and lift the sanctions, Mr. Trump said: “We’ll be looking at that. Yeah, we’ll be looking.”

Mr. Trump’s apparent willingness to avoid condemning Mr. Putin’s government is a remarkable departure from United States policy and Republican Party orthodoxy, and has fueled the questions about Russian meddling in the campaign. Mr. Trump has denied that, saying at the news conference that he has never met Mr. Putin, and has no investments in Russia.

“I would treat Vladimir Putin firmly, but there’s nothing I can think of that I’d rather do than have Russia friendly as opposed to the way they are right now,” he said, “so that we can go and knock out ISIS together.”

Mr. Trump later tried to modify his remarks about hacking Mrs. Clinton’s emails, contending they represented an effort to get the Russians to turn over their trove to the F.B.I.

With the political conventions coming to an end on Thursday, Mr. Trump is expected to receive his first national security briefings from American intelligence agencies in coming days. It is unclear whether those briefings — which describe the global challenges facing the United States but not continuing covert operations or especially sensitive intelligence — will change any of his views.

His comments about Russian hacking came on a day when Obama administration officials were already beginning to develop options for possible retaliation against Russia for the attack on the Democratic National Committee. As is often the case after cyber incidents, the options for responding are limited and can be viewed as seeming too mild or too escalatory.

The administration has not publicly accused the Russian government of the Democratic National Committee hacking, or presented evidence to back up such a case. The leaked documents, first published by a hacker who called himself “Guccifer 2.0” and who is now believed to be a character created by Russian intelligence, portrayed some committee officials as favoring Mrs. Clinton’s candidacy while denigrating her opponent, Senator Bernie Sanders. The release of the internal party emails and documents led to the resignation of Debbie Wasserman Schultz as chairwoman of the party.

Mr. Trump contended on Wednesday that the political uproar over whether Russia was meddling in the election was a “total deflection” from the embarrassing content of the emails. Many Republicans, even some who say they do not support Mr. Trump, say they agree.

If Mr. Trump is serious in his call for Russian hacking or exposing Mrs. Clinton’s emails, he would be urging a power often hostile to the United States to violate American law by breaking into a private computer network. He would also be contradicting the Republican platform, adopted last week in Cleveland, saying that cyberespionage “will not be tolerated,” and promising to “respond in kind and in greater magnitude” to all Chinese and Russian cyberattacks.

In the past, the Obama administration has stopped short of retaliating against Russia — at least in any public fashion — for its attacks on the State Department and White House unclassified email systems, or on networks used by the Joint Chiefs of Staff. It never even publicly identified Russian intelligence as the source of those intrusions, though the subject was widely discussed by senior United States officials when they were not speaking for attribution.

In contrast, the United States did bring indictments against Chinese and Iranian hackers for thefts of intellectual property and attacks on American banks, and imposed economic sanctions against North Korea in early 2015, for hacking into Sony Pictures Entertainment’s computers.

Almost as soon as Mr. Trump spoke, other Republicans raced in to try to reframe his remarks and argue that Russia should be punished. A spokesman for Speaker Paul D. Ryan termed Russia “a global menace led by a devious thug.” The spokesman, Brendan Buck, added: “Putin should stay out of this election.”

Even Gov. Mike Pence of Indiana, Mr. Trump’s running mate, issued a statement, saying that “if it is Russia and they are interfering in our elections, I can assure you both parties and the United States government will ensure there are serious consequences.” Mr. Pence did not attend Wednesday’s news conference because he was giving local television interviews, and an aide to Mr. Pence said that his team had written his statement about Russia before Mr. Trump began speaking.

Shortly after that Mr. Trump sent a message on Twitterdeclaring “If Russia or any other country or person has Hillary Clinton’s 33,000 illegally deleted emails, perhaps they should share them with the FBI!”

The fact that the Democratic committee’s servers were targeted — and, apparently, not those of the Republican National Committee — has brought up inevitable comparisons with the origins of the Watergate scandal, when burglars found little after breaking into Democratic committee offices before the 1972 election. The hackers, more than 40 years later, were more successful: A reconstruction of events suggests the first successful piercing of the Democrats’ networks occurred in June 2015, long before the Russians, or anyone else, could have known Mr. Trump would get the nomination.

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President Trump and Russia's president, Vladimir V. Putin, have exchanged many compliments since the 2016 election. We look at the basis of the mutual respect between the two men. (Updated on July 5, 2017.)Published OnJuly 27, 2016CreditCreditFrom left; Todd Heisler/The New York Times, Yury Kochetkov/European Pressphoto Agency

The Clinton campaign, eager to turn the subject from the chaos caused by the email release to the question of Russian interference, accused Mr. Trump of encouraging Russian espionage.

“This has to be the first time that a major presidential candidate has actively encouraged a foreign power to conduct espionage against his political opponent,” said Jake Sullivan, Mrs. Clinton’s chief foreign policy adviser, whose emails from when he was a State Department aide were among those that were hacked.

“This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue,” he added.

For his part, Mr. Trump cast doubt on the conclusion that Russia was behind the hacking. “I have no idea,” he said. He said the “sad thing” is that “with the genius we have in government, we don’t even know who took the Democratic National Committee emails.”

Mr. Trump then argued that if Russia, or any other foreign government, was behind the hacking, it showed just how little respect other nations had for the current administration.

“President Trump would be so much better for U.S.-Russian relations” than a President Clinton, Mr. Trump said. “I don’t think Putin has any respect whatsoever for Clinton.”

Former Representative Pete Hoekstra of Michigan, a Republican who had served as chairman of the House Intelligence Committee, said Mr. Trump was right to keep hammering Mrs. Clinton on the subject of her private emails.

Mr. Hoekstra said he was untroubled by Mr. Trump’s goading of a foreign power, particularly in light of Mrs. Clinton’s use of a private server while she was secretary of state.

”Trump is bringing up a fairly valid point: Hillary Clinton, with her personal email at the State Department, has put the Russians in a very enviable position,” Mr. Hoekstra said. “Most likely the Russians already have all that info on Hillary.”

But Representative Jason Chaffetz, a Utah Republican who led the House oversight committee that looked into Mrs. Clinton’s emails, was more critical. If Mr. Trump’s comments were meant literally, he said in an interview, “I think he was absolutely wrong and out of line. I would never have said it that way, and I think it was ill-advised.”

If the remark was tongue-in-cheek, he added, it failed at political humor.

Ashley Parker reported from Doral, and David E. Sanger from Washington. Reporting was contributed by Alexander Burns, Amy Chozick and John Harwood from Philadelphia, and Eric Lichtblau from Washington.